March 7, 2022

Life History …

… is the pattern of survival and reproduction.

“pattern” means organized by:

  • Age
  • Size
  • Stage (larva / pupa / adult)

Today mainly:

  • Gotelli - Chapter 3

Survival varies by age!

Fecundity\(^*\) varies by age!<

* - Fecundity is number of offspring per individual over some unit of time.

Monoceros academicus: Three Life Stages

. Larva Sophomore Emeritus
.
Survival 0.5 1 0
Fecundity 0 1.5 0.5
  • Survival is a probability (unitless)
  • Fecundity is an expected number of offspring (n. ind.).

Experiment: results

Change one value ….

Stage Survival Fecundity
1. larvae 0.5 0
2. sophomore 1 2
3. emeritus 0 .5

This time, population grows. And the stable age distribution is a bit different.

Life Table….

A “schedule” of all births and deaths in our population

If you know When and With what probability individuals die and when and how many offspring they have, you can compute:

  • Population age structure:
    • Are there lots of: young individuals? Old individuals? Reproductive age individuals?
  • Life expectancy
  • Population growth rate \(r\) or \(\lambda\)
    • ALl of these models are essentially exponential growth models,
  • Survivorship patterns
    • Does most mortality occur in the very young? The very old? Or equally across all ages?
  • Life history strategy

Often - beyond a description - the real question is:

  • The real question is: what (e.g. what environmental, human, density dependent impact) is influencing these parameters

Life Tables - some “encouragement”

Two main pieces:

  • Fecundity schedules: Number of births per age class - i.e. reproductive contributon of females
  • Survivorship schedules: Number of deaths or probability of death per age class.

The pieces (columns):

Age class \((x)\) Number at Start \((S_x)\) Cumulative Prob. of survival \((l_x = S_x/S_1)\) births/ind \((b_x)\) Surv. prob \((g_x = l_x - l_{x+1})\) Reproductive rate: \((l\times b)\) \(g = lbx\)
1. larva 16 1.0 0 0.5 0 0
2. sophomore 8 0.5 1.5 1 0.75 1.5
3. emeritus 8 0.5 .5 0 0.25 .75
4. beyond 0 0 – – – –
totals: \(\Sigma S_i = 32\) \(\Sigma D_i = S_1 = 16\) \(R_0 = \sum lb = 1\) \(G = {\sum(lbx) \over \sum(lb)} = 2.25\)
  • reproductive rate: \(R_0\) - mean number of offspring / female
  • generation time: \(G\) - mean age of mother

Computing Population Growth

\(x\) \(S_x\) \(l_x\) \(b_x\) \((l\times b)\) \(g = lbx\)
1. larva 16 1.0 0 0.5 0
2. sophomore 8 0.5 1.5 1 0.75
3. emeritus 8 0.5 .5 0 0.25
4. beyond 0 0 – – –
totals: \(\Sigma S_i = 32\) \(R_0 = 1\)
  • reproductive rate: \(R_0\) - mean number of offspring / female
  • generation time: \(G\) - mean age of mother

Growth rate is approximately: \(r \approx \log(R_0)/G\) but that doesn’t work for our example at all

Growth rate exactly solves: \[1 = \sum_{x=1}^n e^{-rx} l(x) b(x)\] Generally this is hard.

In our example:

\[1 = e^{-r} 0 + e^{-2r} {3 \over 4} + e^{-3r} {1\over 4}\]

eventually leads to: \(r = 0\).

Reproductive value

Another important metric: Number of offspring YET to be born.

\[v(x) = {e^{rx}\over l(x)} \sum_{y = x+1}^k e^{-ry} \, l(y)b(y)\]

Survival curves

  • TYPE I: high survivorship for juveniles; most mortality late in life

  • TYPE II: survivorship (or mortality) is relatively constant throughout life

  • TYPE III: low survivorship for juveniles; survivorship high once older ages are reached

Monoceros academicus: Type I

  • TYPE I: high survivorship for juveniles; most mortality late in life. Investment in young and survival. Typical of long-lived species.
Stage Survival Fecundity
1. larvae 0.5 0
2. sophomore 1 2
3. emeritus 0 .5

Pink Salmon (Onchorrhynchus gorbusha)

Jargon:

  • Semelparous: Gives birth (or lays eggs / releases seeds) and dies.
  • Iteroparous: Gives birth many times in life.

Pink Salmon (Onchorrhynchus gorbusha): Type III

  • TYPE III: low survivorship for juveniles; survivorship high once older ages are reached. Basically - produce a whole boatload of offspring and hope for the best. Typically short-lived species.
Stage Survival Fecundity
1. smolt 0.05 0
2. ocean 0.9 0
3. return 0 21

Stable age distributions

Given by:

\[c(x) = {e^{-rx} l(x) \over \sum_{x = 0}^k e^{-rx} l(x)}\]

For M. academicus, pretty easy (since \(r = 0\)), just the \(l(x)\) ratios: 1/2, 1/4, 1/4.

For O. gorbusha, \(r\) is also just about 0 … but is there really a “stable distribution”?

(revisit shiny app: https://egurarie.shinyapps.io/AgeStructuredGrowth/ and see how growth rate affects distribution).

Jargon: if \(r = 0\), we refer to the distribution as “stationary”.

Assumptions …

are the same as exponential growth:

  1. Closed population
  2. No genetic structure (or individual variability)
  3. No time lags
  4. Stationarity:
  • Exp. growth: \(b\) and \(d\) (birth and death rates) constant
  • Stage structured growth: \(l(x)\) and \(b(x)\) schedules are invariant in time.
  1. Some care needs to be taken regarding Age vs Stage

The really interesting ecological questions ask what happens when conditions change

Ways of calculating …

  1. Through time: Cohort analysis - follow a cohort (or several cohorts) through time

  2. Snapshot: Age distribution - almost always something weird turns up!

  3. Mark-racapture, observational analysis of survival and reproduction

If you have N-1 variables you can often infer the missing one!

SO much easier to just use matrices!

… next week we learn ONE EASY TRICK to get both the intrinsic growth rate and stable distribution and reproductive value from a life table using the: